We introduce Cubimorph, a modular interactive device that accommodates touchscreens on each of the six module faces, and that uses a hinge-mounted turntable mechanism to self-reconfigure in the user's hand. Cubimorph contributes toward the vision of programmable matter where interactive devices reconfigure in any shape that can be made out of a chain of cubes in order to fit a myriad of functionalities, e.g. a mobile phone shifting into a console when a user launches a game. We present a design rationale that exposes user requirements to consider when designing homogeneous modular interactive devices. We present our Cubimorph mechanical design, three prototypes demonstrating key aspects (turntable hinges, embedded touchscreens and miniaturization), and an adaptation of the probabilistic roadmap algorithm for the reconfiguration.